This should be plug-n-play, and usually is.
First, try different USB ports. At the same time try different cables.
Second, try the external enclosure on another friend's PC. If still no joy then triple check assembly and wiring in the enclosure. These are usually bulletproof but sometimes mistakes happen.
Third is OS which you didn't mention. The Dell HD -might- have some formatting oddity. But if it workd the last time you used it then itshould still work.
Next try putting HD back in tower to make sure it still works, that's just a troubleshooting back check.
Least likely problem is that the enclosure is faulty. You can test that by trying another HD in it.
The point of all this swapping parts around is to identify the exact point of failure.
LAST GASP: Download a copy of Fedora Live CD, burn it, and boot the laptop into single user mode. Plug in the external enclosure and it should show up by looking at /var/log/messages. If you haven't tried Linux you'll need to learn about mount and cp commands. The reason I mention this is that Linux can *very* often recognise things that other OSes have trouble with.
Oh another solution. Do you have a switch or router around, or can borrow one? Network the two PCs together and move files around that way.
One more. I think I've heard that it's possible to directly connect two Win PCs together with no 'network' as such, they become their own network, but have never tried that.